Effectively there is nothing stored in the Web Shield it is just a scanner, you can have it create a log but that is disabled by default. Obviously if you got an alert by the web shield somehow the last infected: field is populated for that session but on shutdown that isn't retained.
You also have to remember the avast self-defence module should help protect its files so if you were to try and delete any logs (used by the avast log viewer) if you are contemplating there deletion you would first have to disable the self-defence module.
All in all I don't believe there is anything retained in these logs to even contemplate there deletion/clearance, you can get carried away with security, clearing your browser cache, cookies, history, etc are all options which in firefox you can set to remove on closure.